Relative & Expected Frequency

Relative Frequency

Relative frequency

What is relative frequency?

• Relative frequency is an estimate of a probability using results from an experiment

  • For a certain number of trials of that experience, the probability of ‘success’ is:

$\frac{\mathrm{Number} \mathrm{of} \mathrm{successful} \mathrm{outcomes}}{\mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{trials}}$

• If you flip an unfair coin 50 times and it lands on heads 20 times, an estimate for the probability of the coin landing on heads is  $\frac{20}{50}$ (its relative frequency)

  • That is the best estimate we can make, given the data we have

  • We do not know the actual probability

• The more trials that are carried out, the more accurate relative frequency becomes

  • It gets closer and closer to the actual probability 

When will I be asked to use relative frequency?

• Relative frequency is used when theoretical probabilities are unavailable (or not possible to calculate)

  • For example, if you do not know the theoretical probability of being left-handed, you can run an experiment to find an estimate (the relative frequency)

• Sometimes theoretical probabilities are known

  • The theoretical probability of a fair coin landing on heads is 0.5

  • The theoretical probability of a fair standard six-sided dice landing on a six is $\frac{1}{6}$ 

• Relative frequency can be compared to a theoretical probability to test if a situation is fair or biased 

  • If 100 flips of the coin give a relative frequency of 0.48 for landing on heads, the coin is likely to be fair

    • The theoretical probability is 0.5 and 0.48 is close to 0.5

  • If 100 flips of the coin give a relative frequency of 0.13 for landing on heads, the coin is likely to be biased (not fair)

What else do I need to know about relative frequency?

• Relative frequency assumes that there is an equal chance of success on each trial

  • The trials are independent of each other

    • For example, if choosing something out of a bag (a ball, or marble etc), it would need to be replaced each time to use relative frequency

• Any experiments used to calculate relative frequency should be random

  • If the experiment is not random, this could introduce bias

Expected Frequencyvideo

Expected frequency

What is expected frequency?

• Expected frequency refers to the number of times you would expect a particular outcome to occur 

• It is found by multiplying the probability by the number of trials

  • If you flip a fair coin 100 times, you would expect 0.5 × 100 = 50 heads

• Sometimes you need to calculate the relative frequency first

  • If you flip a biased coin 40 times and get 10 heads, how many heads would you expect when flipping 100 times?

    • The relative frequency is $\frac{10}{40}$ = 0.25 from the first experiment

    • 0.25 × 100 = 25, you would expect to get heads 25 times from 100 throws